Gruber P. Convex and Discrete Geometry

56 Convex Bodies
or geometric information about its boundary, or information on its volume, surface
area, diameter, width, etc. Classical tools which sometimes serve this purpose are
support functions, distance and radial functions of a convex body. See, e.g. Schneider
[907]. In the context of algorithmic convex geometry, convex bodies are specified
by various membership oracles, see Sect. 4.2 and, for more information, Grötschel,
Lovász and Schrijver [409]. Here, we consider support functions.
Support Functions and Norms
Let C be a convex body in E d . The position of a support hyperplane HC(u) of C
with given exterior normal vector u �= o is determined by its support function hC :
E d → R defined by
Clearly,
hC(u) = sup{u · y : y ∈ C} for u ∈ E d .
HC(u) = � x : u · x = hC(u) � , H − C (u) = � x : u · x ≤ hC(u) � .
If u ∈ S d−1 , i.e. u is a unit vector, then hC(u) is the signed distance of the origin
o to the support hyperplane HC(u). More precisely, the distance of o to HC(u) is
equal to hC(u) if o ∈ HC(u) − and equal to −hC(u) if o ∈ HC(u) + . Since C is the
intersection of all its support halfspaces by Corollary 4.1, we have
(4) C = � x : u · x ≤ hC(u) for all u ∈ E d�
= � x : u · x ≤ hC(u) for all u ∈ S d−1� .
Assume now that C is a proper convex body in E d with centre at the origin o.We
can assign to C a norm �·�C on E d as follows:
�x�C = inf{λ >0 : x ∈ λC} for x ∈ E d .
The convex body C is the solid unit ball of this new norm on E d .
Without proof, we mention the following: for a proper convex body C with centre
o, its polar body
C ∗ ={y : x · y ≤ 1forx ∈ C}
is also a proper, o-symmetric convex body. A simple proof, which is left to the reader,
shows the following relations between the support functions and the norms corresponding
to C and C ∗ :
hC(u) =�u�C ∗ and hC ∗(u) =�u�C for u ∈ E d .
For more information on polar bodies, see Sect. 9.1.